A fully-discrete virtual element method for the nonstationary Boussinesq equations in stream-function form

In the present work we propose and analyze a fully-coupled virtual element method of high order for solving the two dimensional nonstationary Boussinesq system in terms of the stream-function and temperature fields. The discretization for the spatial variables is based on the coupling C1- and C0-conforming virtual element approaches, while a backward Euler scheme is employed for the temporal variable. Well-posedness and unconditional stability of the fully-discrete problem are provided. Moreover, error estimates in H2- and H1-norms are derived for the stream-function and temperature, respectively. Finally, a set of benchmark tests are reported to confirm the theoretical error bounds and illustrate the behavior of the fully-discrete scheme.